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The critical behavior of infinite families of shift symmetric interactingtheories with higher derivative kinetic terms (non unitary) is considered.Single scalar theories with shift symmetry are classified according to theirupper critical dimensions and studied at the leading non trivial order inperturbation theory. For two infinite families, one with quartic and one withcubic interactions, beta functions, criticality conditions and universalanomalous dimensions are computed. At the order considered, the cubic theoriesenjoy a one loop non renormalization of the vertex, so that the beta functiondepends non trivially only on the anomalous dimension. The trace of the energymomentum tensor is also investigated and it is shown that these two families ofQFTs are conformally invariant at the fixed point of the RG flow.

Scale and Conformal Invariance in Higher Derivative Shift Symmetric Theories